On a Geometric Formula for the Fundamental Solution of Subelliptic Laplacians

Abstract

Abrtract. This paper introduces a new method for constructing fundamental solutions and parametricen for a class of second order subelliptic operators. The method is applied to obtain an explicit fundamental solution for the sublaplacian associated to the hypersurface { Im z2 = (51 Ilk} C C '. The fundamental solution is expressed aa an integral over the characteristic variety of an expression whose denominator is a Hamiltonian action function and whose numerator solves an associated second order transport equation. I . IntroductionWe introduce a new geometric formula for the fundamental solution of subelliptic operators of the form ! j Cj X ; + . . which we then use to construct explicit (local) fundamental solutions for the operators (1.1)is a holomorphic function in the complex plane. We set z = 5 1 + 2x2 and 1 z = = ( X , -2X2) ~ 1991 Mathemoiics Subject Classification. Keyurords and phnaser.c = ( -o o -i t , g -] u ( g + , m -i t ) , with upper and lower directed edges denoted by C+ and C-, respectively. Integrals of vx/g vanish on upper and lower semicircles as their radii increase without bound. Therefore we can integrate on a "dumbbell" with waist at g*. Inside this domain v x / g has a simple pole at g = 0. This yields Math. N h r . 181 (19!)(i) ---2ni c-U(-C,) 1 + e-i*A Jv","" -= -2ni Ci e -i s A / 2 -cos (7rA/2)Kx , n and therefore we obtain which agrees with (3.17) of [GS] and with Theorem 6.2 of PSI.We note that the change of variable T + g is reminiscent of classical calculations in actionangle coordinates. For general holomorphic f(z), AA is not translation invariant with respect to any underlying group structure and we must find KA(z, t;z0,t0) for arbitrary ( z o , t o ) . When (z0,to) # (0,O) we obtain a second invariant of motion, the angular momentum i l (~, z0, 7 ) . The evaluation of formula (1.6) by a residuv calculation shows that g* play a crucial role in the final formula, although g drops out, as it should. These observations help us construct new coordinates from g* and i l k = lirnT-,*m R, which reduce the second order transport equation (1.9) to a hypergeometric differential equation in two variables; this was already solved by APPELI, and PICARD in the 1880's (see [AK]).Here we give one form of the fundamental solution which we shall construct in Section 7.(1.18) Set w = z y + izi. Then for IRe XI < 1 the operator Ax = ZZ + ZZ + A [Z, Z ] has thc realanalytic fundamental solution (1.19) where I%eals/Gaveau/Greiner, Fundamental Solution of Subelliptic Laplacians 85 with (1.22) We already noted that for fundamental solutions one needs g& (and C2*) only, as drops out of the calculations. For other functions of Ax, e.g. the heat kernel, one does need g. In particular Tg and 'ux are universal in the sense that functions f(Ax) of A' given by integral kernels have the following representation (1.23) where f depends only on f. We shall return to these questions in future publications. To put our work in context a bit of history is in order. Subelliptic second order operators arise in a numbe...

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